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A review on Light Extinction Spectrometry as a diagnostic for dust particle characterization in dusty
plasmas
Séverine Barbosa, Fabrice Onofri, L. Couëdel, M. Wozniak, C. Montet, C.
Percé, Cécile Arnas, Laïfa Boufendi, Eva Kovacevic, Johannes Berndt, et al.
To cite this version:
Séverine Barbosa, Fabrice Onofri, L. Couëdel, M. Wozniak, C. Montet, et al.. A review on Light Extinction Spectrometry as a diagnostic for dust particle characterization in dusty plas- mas. Journal of Plasma Physics, Cambridge University Press (CUP), 2016, 82 (4), pp.615820403.
�10.1017/S0022377816000714�. �hal-01457776�
A review on Light Extinction Spectrometry as a diagnostic for dust particle
characterization in dusty plasmas
S. B A R B O S A
1, F. R. A. O N O F R I
1† , L. C O U E D E L
2, M. W O Z N I A K
1⇤, C. M O N T E T
1, C. P E L C E
1, C. A R N A S
2, L. B O U F E N D I
3, E. K O V A C E V I C
3,
J. B E R N D T
3,
A N DC. G R I S O L I A
41 Universit´e Aix-Marseille, CNRS, UMR 7343, IUSTI, 13458 Marseille C´edex 13, FRANCE
2Universit´e Aix-Marseille, CNRS, UMR 7345, PIIM, 13397 Marseille C´edex 20, FRANCE
3 Universit´e d’Orl´eans, CNRS, UMR 6606, GREMI, 45000 Orl´eans, FRANCE
4 CEA, IRFM, F-1308 Saint-Paul-lez-Durance, France
⇤Present affiliation: Institute of Physics, Polish Academy of Sciences, 02-668, Warsaw, Poland (Received ?; revised ?; accepted ?. - To be entered by editorial office)
In this article, a detailed description of the light extinction spectrometry diagnostics is given. It allows the direct in-situ measurement of the particle size distribution and absolute concentration of a dust cloud levitating in plasmas. Using a relatively simple and compact experimental set-up, the dust cloud parameters can be recovered with a good accuracy making minimum assumptions on their physical properties. Special em- phasises are given to the inversion of light extinction spectra and all the required particle shape, refractive index and the light extinction models. The parameter range and the limitations of the diagnostic are discussed. In addition, two examples of measurements in low-pressure gas discharges are presented: in a DC glow discharge in which nanopar- ticles are growing from the sputtering of a tungsten cathode, and in an Argon-Silane radio-frequency discharge.
1. Introduction
The knowledge of nano-(micro-) particle (commonly referred as “dust particles”) pa- rameters in a dusty plasma is essential for a genuine comprehension of the system. In- deed, as the dust particles acquire a net electric charge (usually negative) due to their interactions with the surrounding plasma electrons and ions, the discharge and plasma parameters can be strongly modified. Moreover the forces acting on the dust particles at the origin of their transport are highly dependent on the local plasma properties and thus retroactively depend on the dust size distribution and number density. In low-pressure gas discharges in which nanoparticles are growing (e.g. (Boufendi & Bouchoule 1994;
Samsonov & Goree 1999a,b; Berndtet al.2009; Dominique & Arnas 2007; Kishoret al.
2013; Honget al.2003)), the characterisation of the dust particle cloud is always one of the major issues. In many experiments, the measurements of the particle size distribution (PSD) and number density rely on ex-situ measurements using electron microscopes. For instance, Figure 1 shows electron microscopy images of particles with a wide variety of morphologies: porous and cauliflower-shape particles (Figure 1(a-b)), compact aggregates (Figure 1(c-e)) or even low fractal dimension aggregates (Figure 1(f)).
The evolution of the PSD is recovered by collecting particles for di↵erent discharge
† Email address for correspondence: fabrice.onofri@univ-amu.fr
(a)-(b) (c)-(d) (e) (f)
Figure 1.Electron microscopy images of particle aggregates formed in a DC Argon glow dis- charge: (a) SEM, (b) HR-TEM (Tungsten (Kishoret al.2013)); an Argon-Silane low pressure radio-frequency plasma: (c) TEM, (d) SEM (silicon (Boufendi & Bouchoule 1994)); (e) an aerosol drier (SEM, silicon dioxide (Onofriet al.2013)) and (f) sputtering discharge (SEM, aluminium (Samsonov & Goree 1999b)).
durations. Then, correlations are made with the evolution of the discharge parameters (Wattieaux et al. 2011), such as the self-bias voltage and current harmonics in radio- frequency (RF) discharges, the evolution of the cathode bias in direct-current argon glow discharge and, in many experiments, the plasma light emission. However, techniques relying on ex-situ measurements are very time consuming and not adapted when real- time in-situ knowledge of the PSD and number concentration are required. Recently, real-time evolution of the dust particle size and number concentration was obtained in capacitively-coupled RF discharges by following the electric parameters (RF current and voltage, self-bias) and deducing the modification of the plasma impedance. However, the recovery of the dust particle size and number concentration rely on modelling and a careful calibration of the considered set-up. Moreover, it is restricted to monodisperse spherical particles and cannot properly handle dust particles with complex shapes and polydisperse PSD. Laser light scattering on the dust particles is often used to observe in-situ and in real time the nanoparticle cloud. It is however very difficult to extract the evolutions of the PSD and number concentration from the scattering signals as they de- pend on too many parameters (particle size and shape, number concentration, refractive index) and require observations from many angles. Mie-scattering ellipsometry (Gebauer
& Winter 2003; Hong & Winter 2006; Sebastianet al. 2015) seems to be more accurate diagnostic. It consists in measuring the change of polarisation of a polarised laser-light beam scattered by a cloud of nanoparticles. Coupled with extinction, it allows recovering the PSD and the number concentration. It is however experimentally challenging, as it requires accurate measurements of the ellipsometric angles. The iterative data procedure used to find the particle parameters (PSD, local density, and refractive index) is also not trivial and generally requires strong assumptions making systems with complex shaped particle and polydisperse PSD complicated to handle. Otherwise, a reduction of the num- ber of parameters can be considered. For example, the choice of a model providing the time evolution of the particle radius can help in this way. Dust particle size evolution can also be followed, for instance, by measuring white light scattering at di↵erent angles (Mitic et al.2011). By looking at the ratio of the scattered light at di↵erent angles for given wavelengths, it is possible to recover the particle size and the refractive index. This technique however assumes that the particles are spherical and monodisperse.
In this article, we will focus and review on the basic principle, recent achievements and applications of a diagnostic: the Light Extinction Spectrometry (LES), for the in-situ measurement of the PSD and concentration of dust cloud levitating in plasmas. LES uses the extinction of a collimated broadband light beam to recover the PSD and the absolute concentrations in number and in volume of dust levitating in plasmas. Under appropriate conditions, it has the capability to detect nanoparticles as small as 20 nm,
L
probed zone I ( )0!
Source Detection
particles
I ( )T!
Figure 2.Schematic of the principle of light extinction spectroscopy.
to handle rather complex shaped particles as well as multimodal size distributions. LES is e↵ectively used to measure PSDs and particle concentrations in plasmas, aerosols and colloidal suspensions (Crawleyet al.1997; Tamanaiet al.2006; Rosanvallonet al.2008, 2009; Onofriet al.2009, 2011, 2013; Barbosaet al.2016). The remainder of this paper is organized as follows. In section 2, the bases of the LES are introduced. Special empha- sises are given to the inversion of light extinction signals and all the necessary particle shape, refractive index and light extinction models. Section 3 discusses the parameter range and the limitations of LES. Finally, section 4 gives two examples of measurements in low-pressure gas discharges: a DC glow discharge in which particles are growing from the sputtered from a tungsten cathode, and a CC-RF discharge using a mixture of argon and silane to grow silicon particles.
2. Summary of Light Extinction Spectroscopy (LES) method
2.1. Physical basis and basic equationsLES consists in analyzing the variations of the spectral transmittance of a broadband and collimated light beam passing through the particle cloud to be characterized (e.g.
(Bohren & Hu↵man 1998; Xu 2001; Onofri & Barbosa 2012b)), see Figure 2.
Considering I0 and IT, the spectral intensities of the illuminating and transmitted beams respectively, the light beam spectral transmittanceT( ), also called transmission, is defined as follows:
T( ) = IT( ) Ib( )
I0( ) Ib( ) (2.1)
where is the considered light wavelength andIb accounts for the noise of the detection system (e.g. electronic dark noise of the spectrometer, residual background optical noise of the laboratory and plasma light emission). If the contribution of the light scattered in the forward direction by the dust cloud can be neglected, the measured transmission is simply an exponentially decreasing functionT( ) = exp ( ⌧) of the optical thickness
⌧ = CnCextL of the particle cloud. In the latter relation, Cn stands for the particle concentration in number, L is the length of the probed zone and Cext is an integral quantity representing the particle size and spatially averaged extinction of the particle in the probed zone:
Cext= Z Dmax
Dmin
Cext(D,m)˜ n(D)dD (2.2) whereCextis the extinction cross-section of a single particle with diameterDand complex refractive index ˜m( );Dmin andDmax are the diameter of the smallest and the largest particles respectively, andnthe normalized particle size distribution (PSD) in number.
Cext is the sum of two distinct contributions:Cext=Cabs+Csca, withCabsandSsca the particle absorption and scattering cross-sections, respectively.
For spherical particles,Cextcan be expressed as : Cext=
Z Dmax
Dmin
Qext(D, ,m)˜ ⇡D2
4 n(D)dD (2.3)
whereQext= 4Cext/⇡D2 is the extinction coefficient of a single particle. The concentra- tion in volumeCvand the normalized PSD in volumev can be derived straightfowardly from the corresponding quantities in number:
V(D) =Cvv(D) =⇡Cn
6
Z Dmax
Dmin
D3n(D)dD (2.4)
with :
Z Dmax
Dmin
n(D)dD= Z Dmax
Dmin
v(D)dD= 1 (2.5)
whereV is the PSD in volume from which the PSD in mass can be deduced if the particle density is known. By replacing number quantities by volume ones and by introducing the constant ⇤ = 3L/2 , LES transmission equation can be linearized as follows (Onofri et al.2011):
ln[T( i)] =⇤ Z Dmax
Dmin
Qext( , D,m)˜ V(D)
D dD (2.6)
The equation 2.6 is an inhomogeneous Fredholm equation of the first kind (e.g. (Aster et al.2012; Hansenet al.2012)), which can be discretized to obtain an algebraic relation of the formTb =SV, with:
Tb( i) =ln[T( i)] = XN
j=1
Si,jVj (2.7)
where the quantity to be determined is a vectorV with elementsVj andj= 1,2, ..., N.
Nis the number of size classes (i.e. bins) used to discretize the PSD.Sis aM⇥N matrix called (abusively) the extinction matrix. It is a discrete form of the kernel of the integral equation 2.6. Each of its elementsSi,jrepresents the extinction coefficient divided by the particle diameter of the particle size class j for the wavelength i, with i = 1,2, ...M.
M is the number of wavelengths (or spectral bands) used to discretize the logarithm of the transmittance spectrum (which vector form is noted hereinT, with elementsb Tbi, to simplify the notations). TheM⇥N elements ofSare averaged over the width Dof the size classes and the width of the spectral bands. If D and are constant, a simple midpoint rule can be used for the numerical quadrature (Hansenet al.2012):
Si,j= ⇤
D
Z Dj+ D/2 Dj D/2
Z i+ /2
i /2
Qext( i, Dj,m)˜ V(D)
D d dD (2.8)
This quadrature method works fine until and D are small (i.e. the variations of Qext( , D,m)V˜ (D)/D are negligible over the corresponding ranges). This implies that the dimensions and the condition number of the matrixSremain relatively large, which is not suitable for the inversion step (see§2.2). Depending of the application, it can be necessary to employ quadrature rules using more complex interpolating functions than a mean value (equation 2.8), like the trapezoidal rule (linear fit) or rules based on spline functions (Hansenet al.2012), etc.
Equations 2.1, 2.2 are valid for any particle shape, while equation 2.3 is only valid for spherical particles. As a matter of fact, a parameter like the diameter is not necessarily sufficient, or the unique way, to classify non-spherical particles. It is agreed that the diam- eter of roughly spherical particles refers to the diameter of spherical particles having the same projected cross section, surface, volume or ratio of volume and surface (Mishchenko et al.2000). In that case, the PSD retrieved from LES measurements is well defined and directly comparable with other results (TEM analyses for instance). However, these re- sults may be not appropriate to estimate other properties of the particles (e.g. internal porosity and specific surface). In the case of dense aggregates for instance, it can be more appropriate to classify them in terms of number of monomers or radiuses of gyration (e.g.
see§2.3 and Figure 4).
2.2. Regularization and inversion procedure
To retrieve the PSD and particle concentration from LES measurements, it is necessary to solve the algebraic equationTb =SV. Unfortunately, the direct solutionV= StS 1StTb (where the superscriptstand 1 denote the matrix transpose and inverse operations re- spectively) is not of practical use since we are facing an ill conditioned problem (Hansen 1998; Tikhonov et al. 1995). The condition number of S is always so large that even numerical computation round-o↵errors are enough to strongly disturb the solution. Ex- perimental LES spectra being inevitably contaminated by various noise sources (elec- tronic, residual plasma emission, ect), obtaining a direct solution is unthinkable. Writing equation 2.6 in term of volume quantities rather than number ones helps in minimizing the condition number ofS. However this is still not sufficient to get a stable and reliable solution. The problem must be regularized. In other words, more physical inputs (i.e.a priori) and mathematical constraints must be injected into the problem. The literature on regularization methods is extremely dense and various solutions have been proposed (Tikhonovet al. 1995; Hansen 1994). In the following, we summarized the principle of two methods that were used to investigate particle growth in di↵erent low-pressure gas discharges (see§4 for applications) (Onofriet al. 2011; Barbosaet al.2016).
A regularization method based on a dependant model assumes the shape of the PSD to be determined. This is clearly a strong assumption, which can lead to serious errors if the a priori is too far from the real solution. Conversely, since this method imposes a strong constraint on the solution that is sought, it is rather robust regarding the experimental noise and can be used to directly determine the PSD in number for instance. Such a method can be implemented with a non-negativity-constrained least squares algorithm (since all elements of a PSD are positives, V > 0) and a parametric estimator (also called a performance function) quantifying the quality of the reconstruction (Barbosa et al.2016; Onofriet al.2013). For a two-parameter distribution for instance, the mean diameterD and the corresponding standard deviation D are the two unknowns of the problem. The latter can be determined iteratively using a 2-test for the parametric estimation. In that case, the quantity to minimize may be expressed as:
2 D, D = XN
i=1
Tbi Si,jVj(Dj;D, D)
!(Tbi)
!
(2.9) whereTbi represents an element of the experimental spectrum and the productSi,jVj
the corresponding modeled quantity obtained for the iteration parameters D and D. The statistical weighting coefficients !(Tbi) allow to give a higher weight to the spec- tral bands in which we are more confident. They can be estimated from the analysis of the signal-to-noise ratio (SNR) or, more simply, from the standard deviation of a
set of LES spectra. If not possible, they are set to be equal to unity. Many fittings models can be used a priori, like the Normal, the Log-normal, the Gamma distribu- tion, etc. (Xu 2001; Mishchenko et al. 2000). The Normal distribution is mostly used for quasi-monodisperse particulate medium, while the Log-normal is preferably used for polydisperse systems. As a rule, it is also required to iterate on the limits Dmin and Dmax. This is time consuming and hazardous when there is little knowledge about the particle cloud properties. To overcome this difficulty,DminandDmax are often imposed via the definition of a cut-o↵, or an integral convergence criteria, on the PSD (Onofri et al. 2013; Mishchenko et al. 2000). In the case of the Log-normal distribution, these two parameters can be defined as particle sizes for which the PSD equals 0.1%, i.e.
{Dmin;Dmax|nlnN(DS;µ, )/exp(µ 2) = 0.001}(Onofriet al.2013). When a bimodal distribution is expected for instance, the iteration is performed on the parameters of the two individual distributions,D1, D,1andD2, D,2, plus on the relative weight↵of the two modes:n D;D1, D,1, D2, D,2,↵ =↵n D;D1, D,1 +(1 ↵)n D;D2, D,2 with 0<↵<1 (Barbosaet al. 2016). Note that the two distributions are not necessarily of the same type and they can overlap (see 4.1).
The constrained linear inversion method exists in various forms: Tikhonov regularization (Tikhonov et al. 1995), truncated singular value decomposition (Hansen et al. 2012), Phillips-Twomey (Phillips 1962; Twomey 1963, 1979) methods,... which are all equiv- alent (Hansen 1998). For historical reasons, the PhillipsTwomey method is the most widely used in the field of optical particle characterization (e.g. (Xu 2001; Sentis 2014;
Glasseet al.2015; Wyattet al.1988)). This method allows for more measurements than unknowns (i.e.N < M), but the opposite is also possible (i.e.N > M). With the original Phillips-Twomey , the solution can be directly obtained from a single matrix inversion:
V= StS+ H 1StTb (2.10)
whereHis aM⇥M smoothing matrix and a smoothing parameter (also called Lagrange multiplier). The uncertainties of non-correlated measurements can be taken into account via a diagonal matrix Sw whose elements are, for instance, the quantities 1/ !(Tbi). In the latter case, the solution is obtained from:
V= StS!1S+ H 1StS!1Tb (2.11) The vector (StS!1S+ H) 1provides an estimation of the uncertainties into the solution (Kinget al.1978). Although, equation 2.10 provides an explicit solution to the problem, it is preferable to minimize, in the least square sense, the di↵erence between the measured and the modeled quantities:
V=
⇢ V|Min
V>0
⇢
StS+ H V StTb 2 (2.12)
Depending on the problem and thea priori knowledge available on the expected PSD (e.g sharpness or smoothness) the identity matrix or an approximation to the first or second derivative operator can be chosen for H(Hansen 1994; Glasse et al.2015). The analysis of numerically generated synthetic LES spectra (assumingV, and calculating SV) is used for this purpose. The selection of the smoothing parameter is more diffi- cult. Excessive values of tend to over regularize the solution (which is then excessively low-pass filtered), while too small values of this parameters cannot stabilize the solution (which takes the form of a sum of Dirac-like distributions). Over the di↵erent methods available to estimate the optimal value of the regularization parameter (Asteret al.2012;
Hansen 1998), the L-Curve method (Hansenet al. 2012; Hansen 1994) is certainly the
Rv Rg
n
pr ,mp p
(a) (b) (c) (d)
Figure 3. Numerically generated aggregates. Fractal aggregates composed of 300 monomers with (a)Df = 2.00 and with (b)Df = 2.85 (Onofriet al.2011). Buckyball shaped aggregate of 162 monomers with a regular pentagonal-hexagonal surface lattice and a 3D hexagonal compact internal structure: (c) reflection and (d) transmission images (Onofriet al.2013).
more comprehensive: the norm of the retrieved solution|V |2is plotted versus the resid- ual norm (StS+ H)V StTb 2for a large range of values for . The result is expected to take the form of a L-shaped curve. The optimal value of is the one minimizing at the same time the norm of the retrieved solution and the norm of the residual, i.e. the value associated to the corner of the L-curve obtained (Hansen 1994).
2.3. Particle shape and refractive index models
The particle morphology is a key factor for many obvious reasons, and this is particularly true for LES, since it is an input of the electromagnetic models used to calculate the extinction matrix. As pointed out in the introduction (§1), in reactive plasmas, dust particle clouds are often composed of monomers and their aggregates. To describe these particles, most studies used the spherical shape model. This choice can be justified when monomers are roughly spherical and their aggregates highly compact. It is however more doubtful to choose this particle shape model when nothing is known about the particle or when a more realistic model would involve too many unknowns.
A simple and relatively inexpensive approach to model aggregates of monomers is to use the so-called fractal-aggregate equation: np = kf(Rg/rp)Df (Witten & Sander 1981; Sorensen 2001; Theiler 1990; Wozniak et al. 2012; Wozniak 2012). In the latter power-law equation, np and rp stand for the number and radius of monomers within the aggregate respectively, while kf, Df and Rg are the aggregate prefactor, fractal dimension and radius of gyration (i.e. monomers mean square distance from the centre of mass of the whole aggregate) respectively. There exist several approaches to numerically generate fractal-like aggregates (Theiler 1990; Kaye 1994; Meakin 1983; Jullien & Botet 1987). A tunable Di↵usion Limited Aggregation (DLA) (Witten & Sander 1981) model is probably the most efficient method to produce aggregates with well defined parameters.
A code like DLA 1.13 (Wozniaket al.2012; Wozniak 2012; Mroczkaet al.2012; Wozniak
& Onofri 2012) allows, for instance, the simulation of fractal-aggregates composed of thousands of polydisperse, multi-material and overlapping (i.e. partially melt) monomers.
This code can also provides reflection (e.g. SEM-like) or transmission (e.g. TEM-like) images of these aggregates (Wozniak et al. 2012). As an example, Figure 3(a-b) show for two synthetic aggregates composed of the same number of monomers: 3D rendering views (with POV-Ray (POV-Ray 2010)) and native 2D projections showing the radius of gyration and the radius in volume of the equivalent spheres. The monomers observed in plasma systems are obviously not always spherical and their aggregates not well described by the fractal equation. In that case, it is necessary to develop a dedicated particle shape model using the analysis of SEM/TEM images. It can be based on some dendrite growth
0 25 50 75 100 125 150 0
50 100 150 200 250
np=100
Cross-section radius, DA/2[nm]
Radius of gyration, R
g[nm]
Raw data points (DLA model) Fitting: DA/2=a0+a1Rg+a2Rg2+a3Rg3+a4Rg4 with a0=1.2196; a1=1.4503; a2=-0.0041;
a3=8.6212 10-5; a4=-3.363510-7
(a)
np=3 np=20
0 200 400 600
0 150 300 450 600 750 900
0.6 0.8 (c)1.0
Averaged projection
np=3.8 10-5(DA-81)2.50
Nb. of monomers, np
Raw data Fitting Single projection
(b)
DA
!v(SiO2)=1 for DA"81nm
!v(SiO2)=0.51+4.37/(DA-73.1) for DA> 81nm
Volumefraction,!v
Cross-section diameter, DA[nm]
Figure 4.Numerically generated aggregates - polynomial regression models: (a) evolution of the cross-section radius of aggregates with the same fractal dimension (Df = 2.85) versus their gyration radiuses OAPlum12; (b) evolution of the number of monomers and (c) the volume fraction of buckyball shaped aggregates versus their cross-section diameter (Onofriet al.2013).
mechanisms (Aster et al. 2012), a Gaussian sphere model (Muinonen et al. 1996) or simple geometrical considerations (Onofriet al.2013). Figure 3(c-d) show for instance the reflection and transmission images of a buckyball shaped aggregate exhibiting a regular pentagonal-hexagonal surface lattice (see also Figure 1(e)). The outputs of all particle shape models are also useful to link together the di↵erent parameters of the aggregates (as shown in Figure 4). The inset images in Figure 4(a) represent 2D projections of three particular fractal aggregates over the 25000 generated to obtain a polynomial regression model relating the aggregate cross-section diameter DA = 2p
A/⇡ to their radius of gyration (Onofriet al.2012). In the latter relation,Astands for the statistically averaged 2D projected area of aggregates having the same morphological parameters. Figure 4(b-c) show similar results for buckyball shaped aggregates with, in addition, the evolution of their volume fraction V in monomers (1 V represents the aggregate porosity) (Onofri et al.2013).
The particle complex refractive index ˜mis also a key input of electromagnetic models.
It is a crucial parameter for all light diagnostics, but this is particularly true for LES. In fact, in contrast to multi-angle or ellipsometry techniques (e.g. (Hong & Winter 2006; Xu 2001; Sentiset al.2014)), LES requires the knowledge of the particle material refractive index over a large spectral bandwidth and not only for a single (laser) wavelength. This is clearly one of the main drawbacks of this technique and especially when dealing with multi-component materials such as those produced in fusion plasma devices for instance (Onofriet al. 2009; Sharpeet al. 2002). Di↵erent strategies exist for solving this issue.
Firstly, it is possible to collect in the literature and in on-line databases the refractive index spectra of various materials (i.e. Be, Si, SiC, W, WO3,... ) (Palik 1997; SOPRA 2008; J¨ageret al.2003). This procedure requires somea priori knowledge, or some prior analysis of the composition of particle samples. The particle refractive index is assumed to be identical to the one of the bulk material. Secondly, multi-components materials can be modeled using e↵ective medium approximations like the Maxwell-Garnett or the Bruggeman relations (Bohren & Hu↵man 1998; Bohren 1986). These approximations pro- vide e↵ective electrical permittivities (an thus, the refractive indexes) for materials with embedded inclusions. The latter are expected to be small compared to the wavelength and with a moderate relative refractive index. These approximations, whose validity was
200 300 400 500 600 700 800 0.80
0.85 0.90 0.95 1.00
D=26nm
Spectral transmission,T() [-]!
Wavelength,![nm]
"m(W)=1.00
"m(W)=0.74 with inclusions (vaccum)
"m(W)=0.74 with inclusions ("m(WO
3)=0.26) (a)
200 300 400 500 600 700 800 0.80
0.85 0.90 0.95 1.00
D=65nm
Spectral transmission,T() [-]!
Wavelength,![nm]
"m(W)=1.00
"m(W)=0.74 with inclusions (vaccum)
"m(W)=0.74 with inclusions ("m(WO3)=0.26)
(b)
Figure 5. Numerical prediction with the Lorenz-Mie theory and Maxwell-Garnett e↵ective medium approximation of the transmission of a cloud of monodisperse spherical particles with diameter (a) D = 26 nm , (b) D = 65 nm. Three particle compositions are considered:
pure tungsten (mass fraction, m(W) = 1), porous tungsten (e.g. vacuum or gas inclusions,
m(W) = 0.74), tungsten with tungsten oxide inclusions ( m(W) = 0.74, m(W O3) = 0.26).
The particle concentration is fixed toCn= 1013m 3 for a probing length ofL= 1 m (Barbosa et al.2016).
recently demonstrated (Mishchenko et al. 2014), were used, to evaluate changes in the scattering properties of carbonaceous particles contaminated by an increasing volume fraction of spherical inclusions of tungsten (Onofri et al. 2009), or the influence of the porosity or a partial oxidation of tungsten particles (Barbosaet al.2016). Thirdly, this problem can also be partly handled by the scattering models themselves by deriving the e↵ective refractive index of coated particles (Berndt et al. 2009), aggregates of spheres or electrical dipoles with di↵erent compositions (Onofriet al.2009). Fourthly, the parti- cle refractive index spectra can be determined experimentally by measuring the spectral hemispheric-reflectivity (ideally from deep UV to far infrared) of a particle sample. From these spectra the complex refractive index spectra can be reconstructed using Kramer- sKronig integrals (Ku & Felske 1986). Finally, the PSD can be used as an input of the inversion method that is performed to retrieve the particle refractive index spectra.
2.4. Calculation of the extinction matrix
The calculation of the extinction of the so-called extinction matrix requires an accurate modeling of the particle absorption and scattering cross-sections. For this, as already mentioned, it is necessary to have appropriate particle shape and refractive index models, but also an accurate electromagnetic model. In what follows, a brief review is performed on the principal models relevant for LES ( i.e. particle with size parameters in the range x=⇡D/ : 0.05 30) (Bohren & Hu↵man 1998; Onofri & Barbosa 2012a; Wriedt 1998, 2010).
The Rayleigh-Debye-Gans approximation for fractal aggregates (RDG-FA) is a general- ization of the well-known Rayleigh and Rayleigh-Debye-Gans approximations for opti- cally soft scatterers (x⌘kRg = 1 and|m˜ 1|= 1 , with k= 2⇡/ ). As an approxima- tion, it gives simple analytic expressions for the cross-sections of fractal aggregates, with Cabsa = npCabsp for absorption and Cscaa = n2pCscap g(k, Rg, Df) for scattering, (Guinier et al. 1995; Farias et al. 1996; Dobbins & Megaridis 1991). In the latter relations, the superscriptarefers to the aggregate properties and the sub(super)script prefers to the properties of a single monomer, when g(k, Rg, Df) is a structure factor whose form de-
10 20 30 40 50 10-20
10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12
0.1 0.2 0.3 0.4 0.5
(a)
rp= 20 nm Rg= 163 nm
T-Matrix Caext Caabs Casca RGD-FA
Caext Caabs Casca
Radius of monomers, rp[nm]
np= 20, Df= 1.80;
mp= 1.57+0.56i! "= 532 nm
rp= 20 nm Rg= 81 nm
~
Size parameter, xp
Fractal aggregatecross-sections [m2 ]
10 20 30 40 50
10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12
0.1 0.2 0.3 0.4 0.5
rp=40 nm Rg=399 nm np= 100, Df= 1.80;
mp= 1.57+0.56i!#"= 532 nm.
T-Matrix Caext Caabs Casca (b)
RDG-FA Caext Caabs Casca
Radius of monomers, rp[nm]
rp=20 nm Rg=199 nm
~
Size parameter of monomers, xp
Fractal aggregate cross-sections[m-2 ]
Figure 6. Comparison between RGD-FA and T-Matrix calculations (Wozniak 2012) for the cross-sections of carbonaceous aggregates of the same composition and morphology:
˜
m= 1.56 + 0.57i, = 532 nm,Df = 1.80, for increasing radius of monomers with (a)np= 20 and (b)np= 100.
pends on whether the power-law regime is reached or not (i.e. the aggregate is sufficiently large with respect to the wavelength) (Mroczkaet al.2012). As RDG-FA neglects mul- tiple scattering within the aggregates, its accuracy strongly decreases for high fractal dimension (e.g. compact aggregates like the ones shown in Figure 1(a-e)). Nonetheless, as shown in Figure 6, when compared to a rigorous electromagnetic model, RDG-FA provides reasonably accurate predictions for low fractal dimension aggregates (Wozniak et al.2012; Chakrabarty 2009), e.g Figure 1(f). It is also computationally much more ef- fective than rigorous electromagnetic models, e.g. it takes only few seconds with RDG-FA to get the results reported in Figure 6 when a tens hours are required with the T-Matrix (see below).
The Lorenz-Mie Theory (LMT, e.g. (Bohren & Hu↵man 1998)) solves the basic problem of the scattering of a plane electromagnetic waves by a non-magnetic, isotopic and ho- mogeneous spherical particle (called a ”Mie scatterer”). LMT has been extended during the last decades to account for arbitrary shaped beams as well as multilayered or chiral spheres (Onofri et al. 1995; Yanet al. 2012; Borghese et al. 1994), spheres with inclu- sions (Borgheseet al.1994), spheroids (Asano & Yamamoto 1975), etc. This theory uses a separation variable method to solve Maxwell’s equations with the appropriate bound- ary conditions. Theoretically, it has no limitations in terms of particle size range and refractive index. Nonetheless, due to the difficulty in the calculation of complex func- tions with a large complex argument, the numerical difficulties appears for millimeter sized spherical and cylindrical particles, and a few tens of micrometers for other particle shapes. When coupled with e↵ective medium theories, LMT can handle materials that are heterogeneous at the nanoscale, see Figure 7. In this example, the results obtained with the LMT and the Maxwell-Garnet approximation are in good agreement with rig- orous electromagnetic calculations provided that the monomers (i.e. inclusions) remain much smaller than the illumination wavelength.
The T-Matrix method (T-Matrix)(Waterman 1965) is an integral method coming in two forms, the null field and extended boundary technique method e.g (Wriedt 2007;
Mishchenko & Martin 2013). Its name refers to the calculation of a transformation ma- trix allowing to link, using boundary conditions on a circumscribing sphere, the internal field and scattered field that are expended in terms of spherical vector wave functions.
200 300 400 500 600 700 800 900 1000 1100 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
LMT with MG approx.
T-Matrix
Extinctioncoefficient,Qext[-]
Wavelength,![nm]
DA
Buckyball model
"v(SiO2)= 0.51 0.54, r# p= 40.5nm
np DA[nm] np DA[nm]
12 235.1 162 643.5 42 366.3 252 782.7 92 497.9 362 921.8
(a)
200 300 400 500 600 700 800 900 1000 1100 1E-4
1E-3 0.01 0.1 1
LMT with MG approx.
T-Matrix
Extinction coefficient,Qext[-]
Wavelength,![nm]
DA
Buckyball model
"v= 0.51#0.54, rp= 15.7 nm;
np DA[nm]; np DA[nm]; np DA[nm]
12 46 92 97 252 1 53
42 71 162 125 362 180
(b)
Figure 7.Comparison of the evolution of the extinction coefficient of dense buckyball shaped ag- gregates (of silica nanobeads): Lorenz-Mie theory with Maxwell-Garnett (MG) e↵ective medium approximation versus T-Matrix calculations (Onofriet al.2013).
Once this matrix is determined, all the particle extinction and scattering properties can be deduced. For numerical reasons, depending on the particle refractive index and shape (i.e. aspect ratio notably), this method is still limited to size parameters bellow 10⇠500.
Since it requires large computational resources, its results are mostly used in the form of look-up tables (i.e. Smatrices for various particle shapes and compositions). As an illustration on T-Matrix capabilities (Mackowski & Mishchenko 1996; Mishchenko &
Travis 2010), Figure 8 shows the spectral evolution of the normalized extinction coeffi- cientQaext/Qpext of clouds of amorphous silicon aggregates. In the limit of a low optical thickness of the particle cloud, the latter were simply simulated by averaging the extinc- tion properties of 500 aggregates for each considered case (i.e., multiple scattering e↵ects between di↵erent aggregates is neglected but not multiple scattering e↵ects within the aggregates). In Figure 8(a) the fractal dimension increases fromDf = 1.5 (i.e. chain-like aggregates) toDf = 2.8 (i.e. cauliflower-like aggregates) while the radius of gyration is kept constantRg = 35 nm and the number of monomers increases from 51 to 1000. By contrast, in Figure 8(b) the fractal dimension is kept constant (Df = 2.85) when the number of monomers increases from 11 to 1000 (i.e. Rg increases from 7 to 35 nm). As it can be seen, LES sensitivity to the particle morphology is quite low. This must be considered as positive when little is known on the particle cloud properties.
The Discrete Dipole Approximation (DDA) is a numerical method that solves the prob- lem of the scattering and absorption by an array of polarizable point dipoles in interaction with a monochromatic plane wave (e.g. (Draine & Flatau 1994; Yurkin & Kahnert 2013)).
The particle model, whose shape can be virtually arbitrary, is meshed with thousands or millions of dipoles. The counterpart of this flexibility is that this method requires larger computational facilities. It is also limited in terms of maximum particle size and refractive index (i.e.|m˜|x <0.5⇠1 and|m˜ 1|<2). The limit on the refractive index is the more problematic for LES. In fact, the complex refractive index of most material encountered in plasma devices is rather high in the UV range, precisely where the LES sensitivity to particle morphology is maximum.
2.5. Basic experimental set-up and software requirements
A LES set-up comes in various forms (e.g. (Xu 2001; Onofri & Barbosa 2012b; Glasse et al.2015; Deepak & Box 1978a; Crawleyet al.1997)). To give an illustration of a typical set-up, the authors present below their own diagnostic, see a schematic view on Figure 9.
200 300 400 500 950 1000 0.70
0.75 0.80 0.85 0.90 0.95 1.001.5
Rg= 35 nm; Df= 1.5, 1.6, ..., 2.8, np= 51, 64, 80, 101, 127, 160, 201, 253,
318, 400, 400, 633, 797, 1000 (a)
Spectral transmission,T[!]
Wavelength,"[nm]
2.8 2.0
2.4 Df
200 300 400 500 950 1000
0.70 0.75 0.80 0.85 0.90 0.95 1.00
Spectral transmission,T["]
Wavelength,"[nm]
7
35 21
Df= 2.8;
Rg= 7, 10.5, 14, 17.5, 21, 24.5, 28, 31.5, 35 nm, np= 11, 33, 76, 146, 239, 368, 536, 746, 1000.
Rg (b)
Figure 8.Numerical predictions with the T-Matrix method of the transmission of a cloud of fractal aggregates of amorphous silicon monomers with (a) the same radius of gyration but various fractal dimensions (and numbers of monomers) and (b) the same fractal dimension but various radiuses of gyration (and numbers of monomers). Other parameters are kept constant:
rp= 3.5 nm,kf = 1.593,Cn= 2.1013m 3,L= 1 m (Onofriet al.2011; Wozniak 2012).
Figure 9. Schematic of a typical LES setup implemented on a plasma facility: (1) cathode, (2) grounded anode, (3) glass half cylinders, (4) UV fused silica windows, (5) highly-stabilized Halogen-Deuterium lamp, (6) optical fibers, (7) on-line intensity attenuator, (8) achromatic coupling and collimating optics, (9) parabolic mirrors, (10) diaphragm, (11) optical choppers, (12) UV-NIR spectrometer, (13) computer.
It was developed to study the growth of tungsten and silicon nanoparticles produced in low pressure discharges (Onofriet al.2011; Barbosaet al.2016). This setup has also been used to characterize aerosols of buckyball shaped SiO2nanoagregates (Onofriet al.2013) and aerosols of compact tungsten aggregates in the perspective to develop combined laser induced breakdown spectroscopy (LIBS) and LES measurements in fusion devices (Onofri et al.2012).
This set-up is composed of a highly-stabilized Halogen-Deuterium lamp (a DH-2000- DUV from Ocean Optics), solarisation resistant optical fibers with a 200µm core, an on-line intensity attenuator, achromatic coupling and collimating optics (50 mm focal
length parabolic mirrors), a diaphragm to control the diameter of the probed zone (and thus refine the spatial resolution of the system), a low noise and high dynamic CCD spectrometer (a Maya Pro from Oceans Optics). This spectrometer has an enhanced response in the UV range and a global half-height resolution of 25 nm. Depending of the optical windows installed on the plasma chamber, the e↵ective spectral range of the full system is about = 200 1000 nm. The aperture angle of the detection system is smaller than 0.1o, allowing to neglect the scattering of particles with diameter smaller than⇠2µm. In all situations, and as a prerequisite to calculate the transmittance defined in equation 2.1, the background noise and the reference signal are measured before the plasma ignition. In some situations (long-term experiments, huge plasma light emission, ect.), two optical choppers are used to chop periodically and alternatively the probing beam and the collection optics field of view. This procedure allows measuring successively:
the optical and electronic signal, the plasma light emissions and the measurement signal.
Classically, to further improve the SNR, the final LES spectra is obtained by averaging
⇠20 to⇠200 instantaneous spectra (the minimum exposure time: 6 ms is selected to further improve the SNR). This procedure decreases the nominal spectrometer acquisition rate from⇠160 spectra per second down to⇠2 and even⇠0.5 spectra per second if the choppers are used. All the control of the system and the computations (i.e. spectrometer, choppers, inversion) are performed with a laptop computer.
3. Parameter range and limitations of the technique
It is extremely difficult to define precisely the parameter range and accuracy of LES in terms of particle size and particle concentration ranges. In fact, they depend on many factors: the optical properties of the particle material (and the knowledge we have of them), the probing length, the SNR of the experiment (light source, CCD, plasma emis- sion,...), the stability of the regularization and inversion algorithm, the appropriateness of the electromagnetic light scattering models and particle shape models, etc. Thus, in the following, we restrict ourselves to some general remarks on the expected size range and particle concentration range for a typical LES set-up like the one described in§2.5.
The maximum particle size range of LES is between⇠ min/20 to⇠5 max, i.e.⇠10 nm and⇠5µm for a spectral range of⇠200 1000 nm. However, from our experience, the dynamic of a single measurement can hardly exceedDmax/Dmin ⇠20 30. The lower particle size bound comes from the increasing sensitivity of LES to the particle refractive index (specially the imaginary part) at the expense of the particle size while, correla- tively, the useful part of the spectrum is becoming narrower (see Figure 13(a)). This makes the inversion procedure very unstable. For the upper bound, the main limiting factor is the evolution of the extinction coefficients itself. This coefficient tends toward 2 for large size parameters (Bohren & Hu↵man 1998), making LES spectra lesser sensitive to the particle size. One additional limitation for the upper bound is in the intensity of the forward di↵raction peak, which rapidly increases with the particle size, polluting increasingly the extinction signal. Although, there exist some correction methods to ac- count for the particle scattering (Deepak & Box 1978a,b; Hirleman 1988), the problem becomes rapidly insoluble if LES is not coupled with another technique.
The particle concentration range is limited towards the smaller values by the SNR of the system. Optically, the SNR of LES spectra can only be improved by increasing the probing lengthL. Generally speaking,L can be increased using a multipass optical cell (e.g (Widmannet al.2005))or a resonant optical cavity (e.g. CRDS (Butleret al.2007)).
However, these two classical solutions are difficult to implement on plasma facilities and/or when in-situ measurements are required. In addition, losses in the multipass op-
tical cells tend to rapidly decrease the probing beam intensity, when resonant optical cavities operate on a spectral bandwidth (e.g. a few nanometers or tens nanometers) too narrow to perform LES inversions. Therefore, a better approach for characterizing parti- cles in low concentration is to improve LES signal detection and processing schemes (e.g.
cross-correlation with a multichannel spectrometer), reduce the spectrometer noise (e.g.
cooled CCD), increase LES probing beam energy (e.g. supercontinum light source), etc.
The maximum particle concentration is limited by the particle cloud scattering (single or multiple) (Hirleman 1988). This limit can be extended by decreasing the probing length L (e.g. analysis of the plasma edge), reducing the detection system collection angle in order to performe LES analysis only on ballistic photons (Calbaet al. 2008). As orders of magnitude, based on the experimental conditions reported in §4., the measurement range of LES in term of particle concentrations in volume is found to be in the range of few parts-per-billion to few parts-per-million.
4. Examples of in-situ measurements and comparisons with other techniques
4.1. Characterisation of tungsten nanoparticle growth in a low pressure discharge In this experiment (Barbosa et al. 2016), Tungsten nanoparticles were grown in a DC argon glow discharge initiated between a circular tungsten cathode with a diameter of 10 cm and a stainless-steel grounded anode. A detailed description of the set-up and conditions can be found in (Cou¨edelet al.2014); here only a short summary is proposed.
The inter-electrode distance is 10 cm. Two glasses half-cylinders are used to confine the plasma. A 1 cm gap is kept between them to allow the implementation of optical diag- nostics. A static argon pressure of 0.6 mbar (no gas flow) is set during the experiments.
The electrode assembly is contained in a cylindrical vacuum chamber of 30 cm diameter and 40 cm length. An oil di↵usion pump achieves a base pressure of <10 6 mbar. A manually operated gate valve is used to isolate the chamber from the oil di↵usion pump during experiments. A regulated power supply is used to bias the cathode. The discharge current density is kept at a constant value (0.53 mAcm2 corresponding to a current of 40 mA). The current variations are less than 0.05%. The discharge parameters are such that the plasma mainly consisted of a negative glow. With the aforementioned operating conditions, the cathode is sputtered and tungsten NPs are grown (Kishoret al.2013).
As complementary measurements, electron microscopy and Raman spectroscopy analy- ses were performed (o↵-line) on particles collected on a movable substrate holder. It turns out that the nanoparticles are roughly spherical, and look mainly like compact aggre- gates of tungsten crystallites (see Figure 1(a-b)). These observations as well as numerical simulations with rigorous electromagnetic models, justify the choice for LES inversions of the sphere as the particle shape model and the selection of the Lorenz-Mie theory to calculate the extinction matrix. Since the measured transmissions are quite noisy, due to a relatively low particle concentration, only selected values are used for the inversion procedure (see Figure 10(b)). For the same reason, a dependant model is used to help in the regularization problem. Finally, due to the dynamics of the particle cloud, the PSDs in number and in volume is assumed to be of mono-modal or bi-modal log-normal types (Barbosaet al.2016).
Figure 11 shows, typical time series measured with LES at a height ofh= 2.8 cm for (a) the mean diameters and the associated standard-deviations of each PSD mode and (b) the particle concentrations in number. Note that LES spectra were too noisy before t⇠80 s to allow any reliable inversion. The sudden change of the PSD from mono-modal
200 300 400 500 600 700 800 0.2
0.4 0.985 0.990 0.995 1.000 1.005
(c)
(3) (2) 1
Transmission,T[-]
Wavelength,! [nm]
(1) Full transmission
(2) Selected transmissions (for inversion) (3) Intensity ratio: Iplasma/I0
(b)
Iplasma/I0[-]
4.0 10-4 8.0 10-4
0.0
Excessive plasma emission
(1)
Figure 10.Tungsten particle growth. (a) Schematic of the plasma chamber with the probing zones of LES and the particle collection system. (b) Typical raw spectrum and values selected for the inversion ath= 5.3 cm andt= 100 s. (c) Intensity ratio of the plasma emission collected by LES system with respect to the probing beam intensity.
100 200 300 400 500
10 20 30 40 50 60 70 80 90
(1) (2)! (2)
(1)
Diameters, D[nm]
Time, t [s]
(1) Small mode, D1,"
D,1
(2) Large mode, D2,"
D,2
(a)
100 200 300 400 500
0.01 0.1 1 10
(1) (2)! (3)
(2) Concentrationsinnumber [m-3 ]
Time, t [s]
(1) Small mode, Cn,1 (2) Large mode, Cn,2 (3) Total, Cn (b)
(1)
Figure 11. Tungsten particle growth - LES measurements. (a) Temporal evolution of the mean diameter and associated standard-deviation (represented by bars) of each PSD mode forh= 2.8 cm from the anode. (b) Corresponding evolution for the particle cloud concentration in number (small and large modes, total) (Barbosaet al.2016).
to bi-modal observed in Figure 11 a) is attributed to an agglomeration process. The large mode corresponds to the agglomeration of⇠35 nm particles at constant volume (Bar- bosaet al.2016), when the small mode is constituted by particles of a first generation that are not totally consummated by the agglomeration process and remain in the LES probing zone. Figure 12(a) shows the evolution of the statistical moments of the PSD in number for di↵erent heights above the anode and when the minimum of transmittance is reached (i.e. corresponding to the opening of a “dust free” region in the LES beam, for example at 230 s for h= 2.8 cm and at 170 s for h= 5.3 cm) (Barbosa et al.2016). It can be clearly seen that the agglomeration process and cloud properties are not homo- geneous in the discharge (this was confirmed qualitatively by laser tomography images).
The mean size of the nanoparticles belonging to the large mode increases when getting closer to the anode, while on the contrary, the mean size associated to the small mode shows no clear trend. These result suggest that the agglomeration process is triggered when the biggest nanoparticles fall through a cloud of smaller nanoparticles and grow until they reach the anode side.
Figure 12(b) illustrates the difficulty when comparing in-situ and ex-situ analyses. On one hand, TEM analyses are performed on a sample that is expected to be represen-
2.5 3.0 3.5 4.0 4.5 5.0 5.5 30
35 40 45 50 55 60 65 70
(2)
Diameter, D[nm] (1)
Position above the anode, h [cm]
(1) Small mode, D1,!
D,1
(2) Large mode, D2,!
D,2
( )a
0 100 200 300390392394396398400402404406408 20
40 60 80
Diameter and standard-deviation [nm]
Time, t [s]
LES: D,!
D
(1) Spatial averaging (2) Spatio-temporal averaging TEM: D,!
D
(3) Global spatio-temporal averaging
400 (b)
Figure 12.Tungsten particle growth. (a) Evolution with the height above the anode of the mean diameter and the associated standard-deviation (represented by bars) of each PSD mode, measured with LES at the minima in transmittance. (b) Comparison of TEM statistics with LES statistics (spatially averaged overh= 2.8 toh= 5.3 cm, plus time averaged over the time required to collect the samples).
tative of particle populations within, more or less, a cylindrical volume defined by the particle sample collection system area and the full electrodes inter-distance (as sketched in Figure 10(a)). The sample collection procedure takes time and several discharges are necessary to collect enough particles. Thus, TEM results must be considered as spatio- temporal averaged quantities. On the other hand, LES provides time-resolved statistics representative of the particle cloud properties at a given height. For a better comparison, LES analyses should be performed (and spatially averaged) over the whole electrode gap.
Unfortunately, this was not possible on the current plasma test chamber and indeed, only one third of this distance was accessible for the experiments reported here. Figure 12(b) shows LES global results obtained with these spatial-averaging (for h= 2.8 to 5.3 cm) and time-averaging (over the time required to collect TEM samples, ”spatio-temporal av- eraging” case) procedures. The results are in a qualitative agreement, despite significant fluctuations of the TEM analyses on collected samples. Note that for these mean calcu- lations LES maximum acquisition time was limited by the shortest time series recorded (i.e. 300 s).
4.2. Silicium nanoparticle formation in an argon-silane low pressure radio-frequency plasma
In this experiment (Onofri et al. 2011), silicon nanoparticles were grown in a tran- sient low-pressure argon-silane radio-frequency (RF) discharge where the total pressure
⇠13 mBar, the RF electrode power⇠10 W and the mass fractionYSIH4/Ar= 4% were maintained constant during all the experiment duration. The LES system was installed on both sides of the chamber (Onofriet al.2011) with a probing beam positioned halfway between the electrodes. A detailed description of the setup and conditions can be found.
We summarize here the basic parameters of the plasma test chamber. The discharge is confined in a grounded cylindrical (13 cm diameter) stainless steel box. The shower type upper electrode (driven electrode) is connected to a standard matching box, supplied with Argon and Silane (SiH4, in order to ensure a good homogeneity of the gas mixture at the entrance of the discharge gap. The inter-electrode distance is about 3.3 cm. A grid (50% of transparency) is placed at the bottom of the grounded box in order to obtain a laminar vertical gas flow. The discharge structure is enclosed in a vacuum chamber
